(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 8.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 33582, 990] NotebookOptionsPosition[ 31924, 936] NotebookOutlinePosition[ 32290, 952] CellTagsIndexPosition[ 32247, 949] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Initialization", "Subsection", CellChangeTimes->{{3.544829459525244*^9, 3.544829463091817*^9}}], Cell["Run the following commands to initialize the worksheet", "Text", CellChangeTimes->{{3.544829480858035*^9, 3.544829495465344*^9}}, FontSize->18], Cell[BoxData[ RowBox[{ RowBox[{"rsol2", "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Cos", "[", RowBox[{"\[CapitalOmega]", " ", "t"}], "]"}], SubscriptBox["x", "0"]}], "+", RowBox[{ RowBox[{"Cos", "[", RowBox[{"\[CapitalOmega]", " ", "t"}], "]"}], "t", " ", SubscriptBox["v", "x"]}], "+", RowBox[{ RowBox[{"Sin", "[", RowBox[{"\[CapitalOmega]", " ", "t"}], "]"}], SubscriptBox["y", "0"]}], "+", RowBox[{ RowBox[{"Sin", "[", RowBox[{"\[CapitalOmega]", " ", "t"}], "]"}], "t", " ", SubscriptBox["v", "y"]}], "-", "\[Rho]"}], ",", " ", RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"Sin", "[", RowBox[{"\[CapitalOmega]", " ", "t"}], "]"}]}], SubscriptBox["x", "0"]}], "-", RowBox[{ RowBox[{"Sin", "[", RowBox[{"\[CapitalOmega]", " ", "t"}], "]"}], "t", " ", SubscriptBox["v", "x"]}], "+", RowBox[{ RowBox[{"Cos", "[", RowBox[{"\[CapitalOmega]", " ", "t"}], "]"}], SubscriptBox["y", "0"]}], "+", RowBox[{ RowBox[{"Cos", "[", RowBox[{"\[CapitalOmega]", " ", "t"}], "]"}], "t", " ", SubscriptBox["v", "y"]}]}]}], "}"}]}], ";"}]], "Input", InitializationCell->True, CellChangeTimes->{3.544830865339818*^9}, FontSize->14], Cell[BoxData[{ RowBox[{ RowBox[{"dd", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{"Lighter", "[", RowBox[{"Brown", ",", " ", ".7"}], "]"}], ",", " ", RowBox[{"Disk", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", " ", "0"}], "}"}], ",", " ", "1"}], "]"}]}], "}"}]}], ";"}], "\n", RowBox[{ RowBox[{"\[Rho]", " ", "=", " ", "0"}], ";"}]}], "Input", InitializationCell->True, CellChangeTimes->{{3.544795495899025*^9, 3.544795504029785*^9}, { 3.544795562365148*^9, 3.544795607531466*^9}, {3.544795946534625*^9, 3.544795970472192*^9}, {3.5448111892285624`*^9, 3.544811253249135*^9}, { 3.5448157771843224`*^9, 3.544815780143354*^9}}, FontSize->14], Cell[BoxData[ RowBox[{ RowBox[{"Fix", "[", RowBox[{ "pos_", ",", "vel_", ",", "vframe_", ",", "time_", ",", "steps_", ",", "rot_"}], "]"}], ":=", RowBox[{"Block", "[", RowBox[{ RowBox[{"{", RowBox[{ "G", ",", "tt", ",", "aa", ",", "MM", ",", "rsol", ",", "ll", ",", "ff", ",", "av", ",", "i", ",", "vx", ",", "vy", ",", "rsol3", ",", "t"}], "}"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"If", "[", RowBox[{ RowBox[{ RowBox[{"ToString", "[", "vframe", "]"}], "\[Equal]", "\"\\""}], ",", RowBox[{ RowBox[{"{", RowBox[{"vx", ",", "vy"}], "}"}], "=", RowBox[{"vel", "-", RowBox[{"rot", RowBox[{"{", RowBox[{ RowBox[{ "pos", "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}], ",", RowBox[{"-", RowBox[{ "pos", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}]}], "}"}]}]}]}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{"vx", ",", "vy"}], "}"}], "=", "vel"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"rsol3", "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Cos", "[", RowBox[{"rot", " ", "t"}], "]"}], RowBox[{ "pos", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}], "+", RowBox[{ RowBox[{"Cos", "[", RowBox[{"rot", " ", "t"}], "]"}], "t", " ", "vx"}], "+", RowBox[{ RowBox[{"Sin", "[", RowBox[{"rot", " ", "t"}], "]"}], RowBox[{ "pos", "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}]}], "+", RowBox[{ RowBox[{"Sin", "[", RowBox[{"rot", " ", "t"}], "]"}], "t", " ", "vy"}]}], ",", RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"Sin", "[", RowBox[{"rot", " ", "t"}], "]"}]}], RowBox[{ "pos", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}], "-", RowBox[{ RowBox[{"Sin", "[", RowBox[{"rot", " ", "t"}], "]"}], "t", " ", "vx"}], "+", RowBox[{ RowBox[{"Cos", "[", RowBox[{"rot", " ", "t"}], "]"}], RowBox[{ "pos", "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}]}], "+", RowBox[{ RowBox[{"Cos", "[", RowBox[{"rot", " ", "t"}], "]"}], "t", " ", "vy"}]}]}], "}"}]}], ";", "\[IndentingNewLine]", RowBox[{ SubscriptBox["G", "1"], "=", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{"dd", ",", RowBox[{"Darker", "[", "Green", "]"}], ",", RowBox[{"Arrowheads", "[", "0.05", "]"}], ",", RowBox[{"Thickness", "[", "0.01", "]"}], ",", RowBox[{"Arrow", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "0.25"}], ",", "0"}], "}"}]}], "}"}], "]"}]}], "}"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"Do", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["tt", "i"], ":=", RowBox[{"rot", " ", "i", " ", FractionBox["time", "steps"]}]}], ";", "\[IndentingNewLine]", RowBox[{ SubscriptBox["aa", "i"], ":=", RowBox[{"{", RowBox[{"Blue", ",", RowBox[{"Arrowheads", "[", "0.05", "]"}], ",", RowBox[{"Thickness", "[", "0.01", "]"}], ",", RowBox[{"Arrow", "[", RowBox[{"{", RowBox[{"pos", ",", RowBox[{"pos", "+", RowBox[{ RowBox[{"{", RowBox[{"vx", ",", "vy"}], "}"}], " ", "i", " ", FractionBox["time", "steps"]}]}]}], "}"}], "]"}]}], "}"}]}], ";", "\[IndentingNewLine]", RowBox[{ SubscriptBox["MM", "i"], ":=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", SubscriptBox["tt", "i"], "]"}], ",", RowBox[{"-", RowBox[{"Sin", "[", SubscriptBox["tt", "i"], "]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"Sin", "[", SubscriptBox["tt", "i"], "]"}], ",", RowBox[{"Cos", "[", SubscriptBox["tt", "i"], "]"}]}], "}"}]}], "}"}]}], ";", "\[IndentingNewLine]", RowBox[{ SubscriptBox["rsol", "i"], ":=", RowBox[{ SubscriptBox["MM", "i"], ".", "rsol3"}]}], ";", "\[IndentingNewLine]", RowBox[{ SubscriptBox["ll", "i"], ":=", RowBox[{"ParametricPlot", "[", RowBox[{ SubscriptBox["rsol", "i"], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"i", " ", FractionBox["time", "steps"]}]}], "}"}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{"Red", ",", RowBox[{"Thickness", "[", "0.01", "]"}]}], "}"}]}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{ SubscriptBox["ff", "i"], ":=", RowBox[{"{", RowBox[{ RowBox[{"Darker", "[", "Green", "]"}], ",", RowBox[{"Arrowheads", "[", "0.05", "]"}], ",", RowBox[{"Thickness", "[", "0.01", "]"}], ",", RowBox[{"Arrow", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}], ",", RowBox[{ RowBox[{"-", "0.25"}], RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", SubscriptBox["tt", "i"], "]"}], ",", RowBox[{"Sin", "[", SubscriptBox["tt", "i"], "]"}]}], "}"}]}]}], "}"}], "]"}]}], "}"}]}], ";", "\[IndentingNewLine]", RowBox[{ SubscriptBox["G", RowBox[{"i", "+", "1"}]], "=", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{"dd", ",", RowBox[{ SubscriptBox["ll", "i"], "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], ",", SubscriptBox["aa", "i"], ",", SubscriptBox["ff", "i"]}], "}"}], "]"}]}], ";"}], "}"}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "steps"}], "}"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"Animate", "[", RowBox[{ SubscriptBox["G", "i"], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "steps", ",", "1"}], "}"}]}], "]"}]}]}], "]"}]}]], "Input", InitializationCell->True, CellChangeTimes->{{3.544796219321165*^9, 3.544796300422538*^9}, { 3.544796379717616*^9, 3.544796400326128*^9}, {3.544796462502713*^9, 3.544796550552733*^9}, {3.544796587336793*^9, 3.544796615880436*^9}, { 3.544796698040762*^9, 3.544796698199485*^9}, {3.544796776856007*^9, 3.544796785765395*^9}, {3.544796817573544*^9, 3.5447968193808928`*^9}, { 3.544796880916198*^9, 3.544796881035705*^9}, {3.544805754324964*^9, 3.544805776131587*^9}, {3.5448058250121098`*^9, 3.544805877141198*^9}, { 3.544805955873887*^9, 3.544805983216707*^9}, {3.544810276837427*^9, 3.5448102823712893`*^9}, {3.5448103312531424`*^9, 3.544810498967844*^9}, { 3.544810583766327*^9, 3.5448107354809732`*^9}, {3.544810772536107*^9, 3.544810775252996*^9}, {3.544810816039761*^9, 3.544810935508812*^9}, { 3.544811283888177*^9, 3.544811308945129*^9}, {3.544811367999971*^9, 3.544811386497816*^9}, {3.544811433759869*^9, 3.544811466742241*^9}, { 3.544815719976452*^9, 3.544815720279327*^9}, {3.544815824231366*^9, 3.544816053066422*^9}, {3.544816084874867*^9, 3.5448164500669518`*^9}, { 3.544816783337874*^9, 3.54481680894971*^9}, {3.544817038165185*^9, 3.544817044364925*^9}, {3.5448170945610657`*^9, 3.544817159516139*^9}, { 3.5448173419523163`*^9, 3.544817414244227*^9}, {3.5448174573124743`*^9, 3.544817481319303*^9}, {3.5448175466330757`*^9, 3.544817600267619*^9}, { 3.544817683692074*^9, 3.54481771804205*^9}, {3.5448177845502377`*^9, 3.544817803685011*^9}, {3.544827886235811*^9, 3.54482801385949*^9}, { 3.54482808627596*^9, 3.544828155744339*^9}, {3.544828205722399*^9, 3.544828214941681*^9}, {3.544828245629935*^9, 3.544828373605002*^9}, { 3.544828449230751*^9, 3.544828579738361*^9}, {3.544828628818826*^9, 3.544828744002999*^9}, {3.544828777766114*^9, 3.544828835412543*^9}, { 3.544828880841695*^9, 3.544828881025762*^9}, {3.544829054889943*^9, 3.544829055586479*^9}, {3.544829124968069*^9, 3.544829133503565*^9}, 3.544829183649199*^9, {3.54483614909884*^9, 3.544836163866514*^9}}, FontSize->14], Cell[BoxData[ RowBox[{ RowBox[{"Rot", "[", RowBox[{ "pos_", ",", "vel_", ",", "vframe_", ",", "time_", ",", "steps_", ",", "rot_"}], "]"}], ":=", RowBox[{"Block", "[", RowBox[{ RowBox[{"{", RowBox[{ "H", ",", "f2", ",", "t2", ",", "M2", ",", "l2", ",", "Start2", ",", "v2", ",", "a2", ",", "av", ",", "i", ",", "vx", ",", "vy", ",", "rsol3", ",", "t"}], "}"}], ",", RowBox[{ RowBox[{"If", "[", RowBox[{ RowBox[{ RowBox[{"ToString", "[", "vframe", "]"}], "\[Equal]", "\"\\""}], ",", RowBox[{ RowBox[{"{", RowBox[{"vx", ",", "vy"}], "}"}], "=", RowBox[{"vel", "-", RowBox[{"rot", " ", RowBox[{"{", RowBox[{ RowBox[{"pos", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"-", RowBox[{"pos", "[", RowBox[{"[", "1", "]"}], "]"}]}]}], "}"}]}]}]}], ",", RowBox[{ RowBox[{"{", RowBox[{"vx", ",", "vy"}], "}"}], "=", "vel"}]}], "]"}], ";", "\n", RowBox[{"rsol3", "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Cos", "[", RowBox[{"rot", " ", "t"}], "]"}], " ", RowBox[{"pos", "[", RowBox[{"[", "1", "]"}], "]"}]}], "+", RowBox[{ RowBox[{"Cos", "[", RowBox[{"rot", " ", "t"}], "]"}], " ", "t", " ", "vx"}], "+", RowBox[{ RowBox[{"Sin", "[", RowBox[{"rot", " ", "t"}], "]"}], " ", RowBox[{"pos", "[", RowBox[{"[", "2", "]"}], "]"}]}], "+", RowBox[{ RowBox[{"Sin", "[", RowBox[{"rot", " ", "t"}], "]"}], " ", "t", " ", "vy"}]}], ",", RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"Sin", "[", RowBox[{"rot", " ", "t"}], "]"}]}], " ", RowBox[{"pos", "[", RowBox[{"[", "1", "]"}], "]"}]}], "-", RowBox[{ RowBox[{"Sin", "[", RowBox[{"rot", " ", "t"}], "]"}], " ", "t", " ", "vx"}], "+", RowBox[{ RowBox[{"Cos", "[", RowBox[{"rot", " ", "t"}], "]"}], " ", RowBox[{"pos", "[", RowBox[{"[", "2", "]"}], "]"}]}], "+", RowBox[{ RowBox[{"Cos", "[", RowBox[{"rot", " ", "t"}], "]"}], " ", "t", " ", "vy"}]}]}], "}"}]}], ";", RowBox[{ SubscriptBox["H", "1"], "=", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{"dd", ",", RowBox[{"Darker", "[", "Green", "]"}], ",", RowBox[{"Arrowheads", "[", "0.05", "]"}], ",", RowBox[{"Thickness", "[", "0.01", "]"}], ",", RowBox[{"Arrow", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "0.25"}], ",", "0"}], "}"}]}], "}"}], "]"}]}], "}"}], "]"}]}], ";", RowBox[{"f2", "=", RowBox[{"{", RowBox[{ RowBox[{"Darker", "[", "Green", "]"}], ",", RowBox[{"Arrowheads", "[", "0.05", "]"}], ",", RowBox[{"Thickness", "[", "0.01", "]"}], ",", RowBox[{"Arrow", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "0.25"}], ",", "0"}], "}"}]}], "}"}], "]"}]}], "}"}]}], ";", "\[IndentingNewLine]", RowBox[{"Do", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["t2", "i"], "=", RowBox[{"rot", "*", "i", "*", FractionBox["time", "steps"]}]}], ";", RowBox[{ SubscriptBox["M2", "i"], "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", SubscriptBox["t2", "i"], "]"}], ",", RowBox[{"Sin", "[", SubscriptBox["t2", "i"], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", RowBox[{"Sin", "[", SubscriptBox["t2", "i"], "]"}]}], ",", RowBox[{"Cos", "[", SubscriptBox["t2", "i"], "]"}]}], "}"}]}], "}"}]}], ";", RowBox[{ SubscriptBox["l2", "i"], "=", RowBox[{"ParametricPlot", "[", RowBox[{"rsol3", ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"i", " ", FractionBox["time", "steps"]}]}], "}"}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{"Red", ",", RowBox[{"Thickness", "[", "0.01", "]"}]}], "}"}]}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{ SubscriptBox["Start2", "i"], "=", RowBox[{ SubscriptBox["M2", "i"], ".", "pos"}]}], ";", "\[IndentingNewLine]", RowBox[{ SubscriptBox["v2", "i"], "=", RowBox[{ RowBox[{ SubscriptBox["M2", "i"], ".", RowBox[{"{", RowBox[{"vx", ",", "vy"}], "}"}]}], " ", "i", " ", FractionBox["time", "steps"]}]}], ";", "\[IndentingNewLine]", RowBox[{ SubscriptBox["a2", "i"], "=", RowBox[{"{", RowBox[{"Blue", ",", RowBox[{"Arrowheads", "[", "0.05", "]"}], ",", RowBox[{"Thickness", "[", "0.01", "]"}], ",", RowBox[{"Arrow", "[", RowBox[{"{", RowBox[{ SubscriptBox["Start2", "i"], ",", RowBox[{ SubscriptBox["Start2", "i"], "+", SubscriptBox["v2", "i"]}]}], "}"}], "]"}]}], "}"}]}], ";", RowBox[{ SubscriptBox["H", RowBox[{"i", "+", "1"}]], "=", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{"dd", ",", RowBox[{ SubscriptBox["l2", "i"], "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], ",", SubscriptBox["a2", "i"], ",", "f2"}], "}"}], "]"}]}], ";"}], "}"}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "steps"}], "}"}]}], "]"}], ";", RowBox[{"Animate", "[", RowBox[{ SubscriptBox["H", "i"], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "steps", ",", "1"}], "}"}]}], "]"}]}]}], "]"}]}]], "Input", InitializationCell->True, CellChangeTimes->{{3.544832857415906*^9, 3.54483288637629*^9}, { 3.544832955356067*^9, 3.544833055593374*^9}, {3.544833088546146*^9, 3.544833108351575*^9}, {3.544833156679853*^9, 3.544833269025609*^9}, { 3.544833300379894*^9, 3.544833342838983*^9}, {3.544833384784147*^9, 3.544833724665287*^9}, {3.544833845423542*^9, 3.544833887945712*^9}, { 3.544833919673257*^9, 3.544833942123984*^9}, {3.544833984851469*^9, 3.544834118990269*^9}, {3.54483425142161*^9, 3.544834255233665*^9}, { 3.544834606746572*^9, 3.544834612274192*^9}, 3.544835004521506*^9, { 3.544835128980602*^9, 3.544835144663124*^9}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell["Getting started", "Subsection", CellChangeTimes->{{3.544829553143009*^9, 3.544829555070428*^9}}], Cell["\<\ Straight line motion in the plane traces out a complicated path as seen by a \ rotating observer.The needle of a phonograph moves radially towards the \ center of the record, yet traces out a long spiral on the record! Replace the record by a blank sheet of paper, and the phonograph needle by a \ red pen.Furthermore, allow the needle to start anywhere, and move in any \ fixed direction with constant speed.What curve is drawn on the paper?\ \>", "Text", CellChangeTimes->{{3.544829804826235*^9, 3.544829807008299*^9}}, FontSize->18], Cell["We showed in class that the equation of this curve is", "Text", CellChangeTimes->{{3.544829922808642*^9, 3.544829928183161*^9}}, FontSize->18], Cell[BoxData[ RowBox[{ SubscriptBox["r", "rel"], "=", RowBox[{ RowBox[{ RowBox[{"rsol2", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], " ", StyleBox[ OverscriptBox["r", "^"], FontSlant->"Italic"]}], StyleBox["+", FontSlant->"Plain"], RowBox[{ StyleBox[ RowBox[{"rsol2", "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}], FontSlant->"Plain"], OverscriptBox["\[Theta]", "^"]}]}]}]], "Input", CellChangeTimes->{{3.54482999664619*^9, 3.544830073706303*^9}, { 3.54483018431747*^9, 3.544830244476553*^9}, 3.544830281871361*^9, 3.54483054307547*^9}], Cell["where the initial position of the pen is", "Text", CellChangeTimes->{{3.544829937310249*^9, 3.544829940733488*^9}}, FontSize->18], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["r", "0"], "=", RowBox[{"{", RowBox[{ SubscriptBox["x", "0"], ",", SubscriptBox["y", "0"]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.54483026105501*^9, 3.544830278996624*^9}, 3.544830553668398*^9, 3.544830897637146*^9}], Cell["and its initial velocity is", "Text", CellChangeTimes->{{3.544829946788728*^9, 3.544829951051803*^9}}, FontSize->18], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["v", "fix"], "=", RowBox[{"{", RowBox[{ SubscriptBox["v", "x"], ",", SubscriptBox["v", "y"]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.544830291273893*^9, 3.544830299999914*^9}, 3.544830555028457*^9, 3.544830904799929*^9}], Cell[TextData[{ "Note that ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["v", "x"], " "}], TraditionalForm]], FormatType->"TraditionalForm"], "and ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["v", "y"], " "}], TraditionalForm]], FormatType->"TraditionalForm"], "denote the initial velocity with respect to the ", StyleBox["nonrotating", FontSlant->"Italic"], " frame. The initial velocity in the ", StyleBox["rotating", FontSlant->"Italic"], " frame is given by" }], "Text", CellChangeTimes->{{3.544830330716137*^9, 3.544830443658824*^9}}, FontSize->18], Cell[BoxData[ RowBox[{ SubscriptBox["v", "rel"], "=", RowBox[{ RowBox[{"D", "[", RowBox[{ SubscriptBox["r", "rel"], ",", "t"}], "]"}], "/.", RowBox[{"t", "\[Rule]", "0"}]}]}]], "Input", CellChangeTimes->{{3.54483045645796*^9, 3.544830478497989*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Demos", "Subsection", CellChangeTimes->{{3.544830593477348*^9, 3.544830594068409*^9}}], Cell[TextData[{ "The following demos graph the path of the pen, both as seen by the \ nonrotating observer (blue vector), and as seen by the rotating observer (red \ line).\n\nFirst, we define some parameters. The angular velocity of the \ record (in radians/second counterclockwise) is given by \[CapitalOmega]; a \ typical value is ", Cell[BoxData[ FormBox[ RowBox[{"\[CapitalOmega]", "=", "1"}], TraditionalForm]], FormatType->"TraditionalForm"], ". The number of seconds the demo will run is ", Cell[BoxData[ FormBox["T", TraditionalForm]], FormatType->"TraditionalForm"], ", a typical value is ", Cell[BoxData[ FormBox[ RowBox[{"T", "=", "5"}], TraditionalForm]], FormatType->"TraditionalForm"], ". The total number of pictures is given by ", Cell[BoxData[ FormBox["n", TraditionalForm]], FormatType->"TraditionalForm"], ", so that the time between pictures is ", Cell[BoxData[ FormBox[ RowBox[{"T", "/", "n"}], TraditionalForm]], FormatType->"TraditionalForm"], ". A reasonable choice for ", Cell[BoxData[ FormBox["n", TraditionalForm]], FormatType->"TraditionalForm"], " is 25." }], "Text", CellChangeTimes->{{3.544830963542501*^9, 3.544831123043777*^9}, { 3.544831959540208*^9, 3.544831969710429*^9}, {3.544836354359875*^9, 3.544836354891115*^9}}, FontSize->18], Cell[TextData[{ "(A smoother animation is obtained for larger values of ", Cell[BoxData[ FormBox["n", TraditionalForm]], FormatType->"TraditionalForm"], "; ", Cell[BoxData[ FormBox[ RowBox[{"n", "=", "25"}], TraditionalForm]], FormatType->"TraditionalForm"], " works fairly well. But large values of ", Cell[BoxData[ FormBox["n", TraditionalForm]], FormatType->"TraditionalForm"], " may also take significantly longer to run. See what works well for you.)" }], "Text", CellChangeTimes->{{3.544831141819941*^9, 3.544831197912505*^9}}, FontSize->18], Cell["\<\ In the figures you will construct, the blue arrow represents the linear \ motion seen by an inertial observer, while the red curve represents the \ motion seen by the rotating observer. (The green arrow is to help you keep \ track of the rotation.) Think of the tip of the blue arrow as containing a \ red pen which writes on the record as the record rotates beneath the (moving) \ pen.\ \>", "Text", CellChangeTimes->{{3.544831261588762*^9, 3.544831266394885*^9}}, FontSize->18], Cell[TextData[{ "The demo consists of two functions, one from the point of view of each \ reference frame. The first of these shows what the nonrotating observer \ sees, and takes the form\n\t\t\tFix(", StyleBox["start", FontSlant->"Italic"], ", ", StyleBox["velocity", FontSlant->"Italic"], ", ", StyleBox["vframe", FontSlant->"Italic"], ", ", StyleBox["T", FontSlant->"Italic"], ", ", StyleBox["n", FontSlant->"Italic"], ", \[CapitalOmega])\nHere, ", Cell[BoxData[ FormBox["n", TraditionalForm]], FormatType->"TraditionalForm"], ", ", Cell[BoxData[ FormBox["T", TraditionalForm]], FormatType->"TraditionalForm"], ", \[CapitalOmega] are as defined above, ", StyleBox["start", FontSlant->"Italic"], " denotes the initial position, ", StyleBox["velocity", FontSlant->"Italic"], " denotes the initial velocity, and ", StyleBox["vframe", FontSlant->"Italic"], " specifies whether the initial velocity is with respect to the fixed \ observer (", StyleBox["vframe=FIX", FontFamily->"Courier", FontWeight->"Bold"], ") or the rotating observer (", StyleBox["vframe=ROT", FontFamily->"Courier", FontWeight->"Bold"], "). " }], "Text", CellChangeTimes->{{3.544831294606337*^9, 3.544831299197339*^9}, { 3.544831357796451*^9, 3.544831451857095*^9}, {3.544831696010558*^9, 3.544831809316799*^9}}, FontSize->18], Cell["\<\ Try the following example, which is roughly what happens when you play a \ record.\ \>", "Text", CellChangeTimes->{{3.544831820087044*^9, 3.544831822286449*^9}, { 3.544832128313653*^9, 3.544832156112346*^9}}, FontSize->18], Cell[BoxData[ RowBox[{"Fix", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", " ", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"1", "/", "4"}], ",", " ", "0"}], "}"}], ",", " ", "FIX", ",", " ", "4", ",", " ", "25", ",", RowBox[{"-", "10"}]}], "]"}]], "Input", CellChangeTimes->{{3.544816554695325*^9, 3.544816608596518*^9}, { 3.544816826898573*^9, 3.544816827009817*^9}, {3.544831890207778*^9, 3.5448318972988*^9}}], Cell[TextData[{ "To see this from the reference frame of the rotating observer, use the \ command\n\t\t\tRot(", StyleBox["start", FontSlant->"Italic"], ", ", StyleBox["velocity", FontSlant->"Italic"], ", ", StyleBox["vframe", FontSlant->"Italic"], ", ", StyleBox["T", FontSlant->"Italic"], ", ", StyleBox["n", FontSlant->"Italic"], ", \[CapitalOmega])\nwhich takes the same parameters. Compare your previous \ output with" }], "Text", CellChangeTimes->{{3.54483204918953*^9, 3.544832103941342*^9}}, FontSize->18], Cell[BoxData[ RowBox[{"Rot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", " ", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"1", "/", "4"}], ",", " ", "0"}], "}"}], ",", " ", "FIX", ",", " ", "4", ",", " ", "25", ",", RowBox[{"-", "10"}]}], "]"}]], "Input", CellChangeTimes->{{3.544816554695325*^9, 3.544816608596518*^9}, { 3.544816826898573*^9, 3.544816827009817*^9}, {3.544831890207778*^9, 3.5448318972988*^9}, {3.544832192234474*^9, 3.544832192605749*^9}}], Cell[TextData[{ "You can reproduce some interesting special cases (Figure 10-4 on page 395 \ of Thornton and Marion) using ", Cell[BoxData[ FormBox[ RowBox[{"\[CapitalOmega]", "=", "1"}], TraditionalForm]], FormatType->"TraditionalForm"], ", the initial position" }], "Text", CellChangeTimes->{{3.544832221115917*^9, 3.544832273851925*^9}}, FontSize->18], Cell[BoxData[ RowBox[{ RowBox[{"MTstart", "=", RowBox[{"{", RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], ",", "0"}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.544832279388794*^9, 3.544832292002813*^9}}], Cell[TextData[{ "one of the initial velocities (and total time ", Cell[BoxData[ FormBox["T", TraditionalForm]], FormatType->"TraditionalForm"], ") below:" }], "Text", CellChangeTimes->{{3.544832306654714*^9, 3.544832339422658*^9}}, FontSize->18], Cell[BoxData[{ RowBox[{ RowBox[{ SubscriptBox["MTv", "a"], "=", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}], FractionBox["3", "2"]}]}], ";", " ", RowBox[{ SubscriptBox["T", "a"], "=", "0.86"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ SubscriptBox["MTv", "b"], "=", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}], FractionBox["4", "5"]}]}], ";", " ", RowBox[{ SubscriptBox["T", "b"], "=", "2.9"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ SubscriptBox["MTv", "c"], "=", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}], FractionBox["9", "20"]}]}], ";", " ", RowBox[{ SubscriptBox["T", "c"], "=", "17.3"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ SubscriptBox["MTv", "d"], "=", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}], "0.328"}]}], ";", " ", RowBox[{ SubscriptBox["T", "d"], "=", "5.0"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ SubscriptBox["MTv", "e"], "=", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "1"}], "}"}], FractionBox["0.47", SqrtBox["2"]]}]}], ";", " ", RowBox[{ SubscriptBox["T", "e"], "=", "3.83"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ SubscriptBox["MTv", "f"], "=", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "1"}], "}"}], FractionBox["0.283", SqrtBox["2"]]}]}], ";", " ", RowBox[{ SubscriptBox["T", "f"], "=", "3.3"}], ";"}], "\[IndentingNewLine]"}], "Input", CellChangeTimes->{{3.544832746219487*^9, 3.544832746593078*^9}, 3.544835334283401*^9, {3.544835364678686*^9, 3.544835411562396*^9}, { 3.544835472953002*^9, 3.544835488131612*^9}, {3.544835827228169*^9, 3.544835829272929*^9}}], Cell[TextData[{ "and specifying ", StyleBox["ROT", FontFamily->"Courier", FontWeight->"Bold"], " for ", StyleBox["vframe", FontFamily->"Courier", FontWeight->"Bold"], ". Try it! Try other choices!" }], "Text", CellChangeTimes->{{3.544832360559019*^9, 3.544832375612013*^9}}, FontSize->18], Cell[BoxData[ RowBox[{"Fix", "[", RowBox[{"MTstart", ",", SubscriptBox["MTv", "f"], ",", "ROT", ",", SubscriptBox["T", "f"], ",", "25", ",", "1"}], "]"}]], "Input", CellChangeTimes->{{3.544835177178318*^9, 3.544835203924323*^9}, { 3.544835608935212*^9, 3.544835647760349*^9}, {3.544835915340284*^9, 3.544835933321474*^9}, {3.544835991287108*^9, 3.544835994512055*^9}, { 3.544836231498348*^9, 3.544836256794744*^9}, {3.544880468413476*^9, 3.544880468518034*^9}}] }, Closed]] }, AutoGeneratedPackage->None, WindowSize->{810, 682}, WindowMargins->{{Automatic, 104}, {16, Automatic}}, FrontEndVersion->"7.0 for Linux x86 (32-bit) (February 25, 2009)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[579, 22, 102, 1, 65, "Subsection"], Cell[684, 25, 151, 2, 43, "Text"], Cell[838, 29, 1397, 42, 69, "Input", InitializationCell->True], Cell[2238, 73, 712, 19, 67, "Input", InitializationCell->True], Cell[2953, 94, 9239, 228, 610, "Input", InitializationCell->True], Cell[12195, 324, 7348, 198, 626, "Input", InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell[19580, 527, 103, 1, 51, "Subsection"], Cell[19686, 530, 547, 9, 183, "Text"], Cell[20236, 541, 150, 2, 43, "Text"], Cell[20389, 545, 629, 19, 54, "Input"], Cell[21021, 566, 137, 2, 43, "Text"], Cell[21161, 570, 302, 9, 54, "Input"], Cell[21466, 581, 124, 2, 43, "Text"], Cell[21593, 585, 305, 9, 54, "Input"], Cell[21901, 596, 596, 22, 74, "Text"], Cell[22500, 620, 273, 8, 80, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[22810, 633, 93, 1, 51, "Subsection"], Cell[22906, 636, 1327, 37, 239, "Text"], Cell[24236, 675, 573, 17, 99, "Text"], Cell[24812, 694, 493, 9, 155, "Text"], Cell[25308, 705, 1362, 49, 241, "Text"], Cell[26673, 756, 236, 6, 71, "Text"], Cell[26912, 764, 483, 13, 54, "Input"], Cell[27398, 779, 536, 21, 99, "Text"], Cell[27937, 802, 529, 13, 54, "Input"], Cell[28469, 817, 366, 10, 71, "Text"], Cell[28838, 829, 234, 7, 78, "Input"], Cell[29075, 838, 254, 8, 43, "Text"], Cell[29332, 848, 1781, 60, 456, "Input"], Cell[31116, 910, 305, 12, 44, "Text"], Cell[31424, 924, 484, 9, 54, "Input"] }, Closed]] } ] *) (* End of internal cache information *)